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JOURNALS // Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika // Archive

Izv. Vyssh. Uchebn. Zaved. Mat., 2022 Number 8, Pages 81–86 (Mi ivm9804)

This article is cited in 4 papers

Brief communications

Chaotic topological foliations

N. I. Zhukova, G. S. Levin, N. S. Tonysheva

National Research University Higher School of Economics, 25/12 Bol. Pecherskaya str., Nizhny Novgorod, 603155 Russia

Abstract: We call a foliation $(M, F)$ on a manifold $M$ chaotic if it is topologically transitive and the union of closed leaves is dense in $M$. The chaotic topological foliations of arbitrary codimension on $n$-dimensional manifold can be considered as multidimensional generalization of chaotic dynamical systems in the sense of Devaney. For topological foliations covered by fibrations we prove that a foliation is chaotic if and only if its global holonomy group is chaotic. Applying the method of suspension, a new countable family of pairwise non isomorphic chaotic topological foliations of codimension two on $3$-dimensional closed and non closed manifolds is constructed.

Keywords: foliation, chaotic foliation, suspended foliation, global holonomy group.

UDC: 515.16

Received: 17.06.2022
Revised: 17.06.2022
Accepted: 29.06.2022

DOI: 10.26907/0021-3446-2022-8-81-86


 English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2022, 66:8, 66–70


© Steklov Math. Inst. of RAS, 2025